For maximum convenience, easy-to-use interface M-functions are included in the Basic Systems and Control toolbox, explicitly addressing some of supported features. Whenever possible, these M-functions allow to work with system objects defined in the MATLAB Control Toolbox.

The following tables contain lists of the main M-functions for basic computations in linear systems analysis and synthesis:

### Solution of Lyapunov and Sylvester equations

sllyap |
Solution of continuous-time Lyapunov equations |

slstei |
Solution of Stein equations |

slstly |
Solving stable continuous-time Lyapunov equations for the Cholesky factor of the solution |

slstst |
Solving stable Stein equations for the Cholesky factor of the solution |

slsylv |
Solution of Sylvester equations |

sldsyl |
Solution of discrete-time Sylvester equations |

### Solution of generalized Lyapunov and Sylvester equations

slgely |
Solution of generalized continuous-time Lyapunov equations |

slgest |
Solution of generalized Stein equations |

slgsly |
Solving stable generalized continuous-time Lyapunov equations for the Cholesky factor of the solution |

slgsst |
Solving stable generalized Stein equations for the Cholesky factor of the solution |

slgesg |
Solution of generalized Sylvester system of equations |

### Solution of Riccati equations

slcares |
Solution of continuos-time algebraic Riccati equation (CARE) with Schur method |

slcaresc |
Solution of CARE with refined Schur method and condition estimation |

slcaregs |
Solution of CARE with generalized Schur method on an extended matrix pencil |

sldares |
Solution of discrete-time algebraic Riccati equation (DARE) with Schur method |

sldaresc |
Solution of DARE with refined Schur method and condition estimation |

sldaregs |
Solution of DARE with generalized Schur method on an extended matrix pencil |

sldaregsv |
Solution of DARE with generalized Schur method on a symplectic pencil |

### Solution of generalized Riccati equations

slgcare |
Solution of generalized continuos-time algebraic Riccati equation (CARE) with generalized Schur method |

slgdare |
Solution of generalized discrete-time algebraic Riccati equation (DARE) with generalized Schur method |

### Condition estimation for Lyapunov and Riccati equations

lyapcond |
Condition estimation for a Lyapunov equation |

steicond |
Condition estimation for a Stein equation |

carecond |
Condition estimation for a CARE |

darecond |
Condition estimation for a DARE |

### Controllability/observability/minimal realization

slconf |
Controllability staircase form of a system |

slobsf |
Observability staircase form of a system |

slminr |
Minimal realization of a system |

### Similarity transformations

slsbal |
Balancing the system matrix of a state-space system |

slsrsf |
Reduction of the state matrix of a state space system to a real Schur form |

slsorsf |
Reduction of the state matrix of a state space system to an ordered real Schur form |

slsdec |
Additive spectral decomposition of a system with respect to a given stability domain |

### Factorization of transfer matrices

lcf |
Left coprime factorization with prescribed stability degree |

rcf |
Right coprime factorization with prescribed stability degree |

lcfid |
Left coprime factorization with inner denominator |

rcfid |
Right coprime factorization with inner denominator |

### System inter-connections

slosfeed |
Closed-loop system for a mixed output and state feedback control law |

slofeed |
Closed-loop system for an output feedback control law |

### System norms

slH2norm |
H2/L2 norm of a system |

slHknorm |
Hankel-norm of a stable projection of a system |

slinorm |
L-infinity norm of a system |

slstabr |
Complex stability radius |

### Canonical forms and system transformations for descriptor systems

slgconf |
Controllability staircase form |

slgobsf |
Observability staircase form |

slgminr |
Irreducible form of a system |

slgsbal |
Balancing the system matrix of a descriptor system |

slgsHes |
Reduction of the pair (A,E) of a descriptor system to a generalized Hessenberg form |

slgsQRQ |
Transformation to a QR- or RQ-coordinate form |

slgsrsf |
Transformation to a real generalized Schur form |

slgsSVD |
Transformation to a singular value decomposition (SVD) or SVD-like coordinate form |

### Poles and zeros

nrank |
Normal rank of the transfer-function matrix of a standard system |

polzer |
Normal rank, poles, zeros, and the Kronecker structure of the system pencil for a standard or descriptor system |

slpole |
Poles of a standard or descriptor system |

slzero |
Normal rank, zeros, and the Kronecker structure of the system pencil for a standard or descriptor system |

### Kalman filter

convKf |
One recursion of the conventional Kalman filter equations |

srcf |
Combined measurement and time update of one iteration of the time-varying or time-invariant Kalman filter in the Square Root Covariance Form |

srif |
Combined measurement and time update of one iteration of the time-varying or time-invariant Kalman filter in the Square Root Information Form |

### Benchmarks

ctdsx |
Benchmark examples for time-invariant, continuous-time, dynamical systems |

dtdsx |
Benchmark examples for time-invariant, discrete-time, dynamical systems |

ctlex |
Benchmark examples of (generalized) continuous-time Lyapunov equations |

dtlex |
Benchmark examples of (generalized) discrete-time Lyapunov equations |

### Special numerical linear algebra computations

bdiag |
Block diagonalization of a general matrix or a matrix in real Schur form |

persch |
Periodic Hessenberg or periodic Schur decomposition of a matrix product |

Hameig |
Eigenvalues of a Hamiltonian matrix using square-reduced approach |

habalance |
Symplectic scaling of a Hamiltonian matrix |

haconv |
Storage representation conversions for a Hamiltonian matrix |

haeig |
Eigenvalues of a Hamiltonian matrix using HAPACK approach |

hapvl |
Paige-Van Loan's form of a Hamiltonian matrix |

haschord |
Reordering the Schur form of a Hamiltonian matrix |

hastab |
Complete stable/unstable invariant subspace of a Hamiltonian matrix |

hasub |
Selected stable/unstable invariant subspace of a Hamiltonian matrix |

haurv |
Symplectic URV form of a general 2n-by-2n matrix |

haurvps |
Symplectic URV/periodic Schur form of a Hamiltonian matrix |

shbalance |
Symplectic scaling of a skew-Hamiltonian matrix |

shconv |
Storage representation conversions for a skew-Hamiltonian matrix |

sHHeig |
Eigenvalues of skew-Hamiltonian/(skew-)Hamiltonian matrix pencils |

sHHstab |
Complete stable right deflating subspace (and a companion subspace, for pencils in factored form) |

sHHurvps |
Generalized symplectic URV/periodic Schur form of a skew-Hamiltonian/(skew-)Hamiltonian matrix pencil |

TLS |
Solution of the Total Least Squares problem using a singular value decomposition (SVD) approach or a Partial SVD (PSVD) approach |

### Structured matrix factorizations

fstchol |
Factorization of a symmetric positive definite (block) Toeplitz matrix T, and solution of associated linear systems, given the first (block) row / column of T |

fstgen |
Factorization of a symmetric positive definite (block) Toeplitz matrix T, computation of the generator of its inverse, inv(T), and/or solution of associated linear systems using the Cholesky factor of inv(T), given the first (block) row / column of T |

fstsol |
Solution of linear systems X T = B / T X = B, where T is a symmetric positive definite (block) Toeplitz matrix, given the first (block) row / column of T |

fstupd |
Factorization and/or updating a factorization of a symmetric positive definite (block) Toeplitz matrix T, and solution of associated linear systems, given the first (block) row / column of T |

fstqr |
Orthogonal-triangular decomposition of a (block) Toeplitz matrix T and solution of associated linear least-squares problems, given the first (block) row and (block) column of T |

fstlsq |
Solution of the linear least-squares problems min(B - T X) or finding the minimum norm solution of T' Y = C, where T is a (block) Toeplitz matrix with full column rank, given the first (block) column and the first (block) row of T |

fstmul |
Matrix-vector products x = T b for a (block) Toeplitz matrix T, given the first (block) column and the first (block) row of T |

### Data analysis

sincos |
Sine or cosine transform of a real signal |

slHart |
Discrete Hartley transform of a real signal |

slconv |
Convolution of two real signals using either fast Fourier transform (FFT) or Hartley transform |

sldeconv |
Deconvolution of two real signals using either FFT or Hartley transform |

slwindow |
Anti-aliasing window applied to a real signal |

The MEX-functions are more difficult to use than the provided M-functions, but allow a greater flexibility. They are called by the M-functions. The following table contains the list of MEX-files for basic systems analysis and synthesis:

linmeq |
Solution of linear matrix equations |

genleq |
Solution of generalized linear matrix equations |

aresol |
Solution of Riccati equations |

aresolc |
Solution of Riccati equations with condition and forward error bound estimates |

garesol |
Solution of generalized (descriptor) Riccati equations |

arecond |
Condition estimate and forward errors for Lyapunov and algebraic Riccati equations |

arebench |
Benchmark examples for algebraic Riccati equations |

condis |
Continuous-time - discrete-time bilinear transformation |

sysconn |
Inter-connections of two systems given in state-space |

sysfconn |
Closed-loop system corresponding to the output, or mixed output and state, feedback control law |

invert |
Dual or inverse of a linear (descriptor) system |

deadbeat |
Minimum norm feedback matrix performing deadbeat control on a matrix pair |

polass |
Partial pole assignment |

syscom |
Controllability/observability/minimal realization |

systra |
System similarity transformation |

gsyscom |
Controllable, observable, or irreducible form |

gsystra |
Equivalence transformations for descriptor systems |

isprpr |
Properness test for the transfer function of a descriptor system |

specfact |
Spectral factorization of a real polynomial |

cfsys |
Factorization of transfer matrices |

syscf |
State-space representation of a system from the factors of its left or right coprime factorization |

slmexp |
Matrix exponential and its integral |

ldsimt |
Output response of a linear discrete-time system |

Hnorm |
System norms and complex stability radius |

linorm |
L-infinity norm of a system |

polezero |
Normal rank, poles, zeros, and the Kronecker structure of the system pencil for a standard or descriptor system |

polezeroz |
Normal rank, poles, zeros, and the Kronecker structure of the system pencil for a standard or descriptor system with complex matrices |

Kfiltupd |
Combined measurement and time update of one iteration of the Kalman filter |

bldiag |
Block-diagonalization of a matrix |

perschur |
Periodic Hessenberg or periodic Schur decomposition of a matrix product |

Hamileig |
Eigenvalues of a Hamiltonian matrix using the square-reduced approach |

hapack_haeig |
Eigenvalues of a Hamiltonian matrix using the HAPACK approach |

HaeigZ |
Eigenvalues of a complex Hamiltonian matrix using the HAPACK approach |

symplURV |
Eigenvalues and generalized symplectic URV decomposition of a skew-Hamiltonian/Hamiltonian pencil in factored form |

symplURVZ |
Eigenvalues and generalized symplectic URV decomposition of a complex skew-Hamiltonian/Hamiltonian pencil in factored form |

skewHamil2eig |
Eigenvalues and orthogonal decomposition of a skew-Hamiltonian/skew-Hamiltonian pencil |

skewHamil2feig |
Eigenvalues and orthogonal decomposition of a skew-Hamiltonian/skew-Hamiltonian pencil in factored form |

skewHamileig |
Eigenvalues and orthogonal decomposition of a skew-Hamiltonian/Hamiltonian pencil |

skewHamileigZ |
Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil |

skewHamildefl |
Eigenvalues of a skew-Hamiltonian/Hamiltonian pencil and the right deflating subspace corresponding to the eigenvalues with strictly negative real part |

skewHamildeflf |
Eigenvalues of a skew-Hamiltonian/Hamiltonian pencil in factored form and the right deflating subspace corresponding to the eigenvalues with strictly negative real part |

skewHamildeflZ |
Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil and the right deflating subspace corresponding to the eigenvalues with strictly negative real part |

skewHamildeflfZ |
Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil in factored form and the right deflating subspace corresponding to the eigenvalues with strictly negative real part |

TotalLS |
Solution of the Total Least Squares problem using a singular value decomposition (SVD) approach or a Partial SVD (PSVD) approach |

fstoep |
Factorization of a symmetric positive definite (block) Toeplitz matrix and/or solution of associated linear systems |

fstoeq |
QR factorization of a (block) Toeplitz matrix and/or solution of associated linear least-squares systems |

datana |
Data analysis |

The MEX-files above provide interfaces to the main user-callable or computational routines for basic systems analysis and synthesis, and cover all functionality available in the corresponding SLICOT routines.

Executable SLICOT MEX-files are provided for recent MATLAB releases running under WINDOWS and Linux. Demonstration packages can also be provided.

For purchasing licenses of SLICOT-based MATLAB Toolboxes, please visit the SynMath homepage of our partner company SynOptio GmbH.

This email address is being protected from spambots. You need JavaScript enabled to view it. March 10, 2005; Updated: December 2, 2012